They should, but that doesn't mean they:

- write nip-65 to as much relays as possible

- have such relays as purplepag.es in their default relay list which gets used for nip-65 discovery.

Aggregators help a bit, but we should rely on them to make this work.

I still prefer that my nip-65 does not get overwritten without my explicit action on some stupid client.

For outbox model and for people you follow that you can't find a relay list for (BTW you should also check kind 3 and nip05 relay lists as a fallback) I would simply use a default list for the outbox calculation, but still wouldn't force a broadcast with a newer timestamp and that specific client chosen default list.